Recently, lithium ion secondary batteries using, as a negative electrode active material, a lithium titanate, for example, represented by a chemical formula of Li4Ti5O12, have been developed. Such a battery has an advantage of higher safety against an internal short-circuit compared to lithium ion secondary batteries using a carbon negative electrode, because it becomes insulator when the lithium titanate is in a discharged state. On the other hand, in the lithium titanate, a reduction reaction, caused by over-insertion of Li, can sometimes be advanced during overcharge. When such a reduction reaction is advanced, a negative electrode potential suddenly drops, and a side-reaction between a negative electrode and a liquid electrolyte suddenly occurs. Such a suddenly occurring side-reaction may cause increase of a temperature in an inside of the battery due to sudden heat generation. For that reason, the over-insertion of Li to the lithium titanate may cause lowered safety of the battery.